Since this isn't exactly what you asked, I'll leave it as a comment. Perhaps you will be interested in the crossed product algebras. Consider a Hopf algebra and some algebra acting on it, then you can form a Hopf algebra with underlying vector space of their tensor product. If you need a reference, I would suggest nlab. Sorry if this is off-topic.
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B. BischofDec 24 '11 at 1:02

Also the tensor and the shuffle Hopf algebra of a vector space (or, better, module). Besides, the Malvenuto-Reutenauer Hopf algebra $\mathrm{FQSym}$ (see math.tamu.edu/~maguiar/MR.pdf for definition and many properties) is a good example of something graded, neither commutative nor cocommutative but still easy to calculate in. There is also the Loday-Ronco Hopf algebra, but I am not really understanding its definition yet; it is definitely not that simple. But at least something made out of trees would be useful. I don't remember how the Hopf algebra in ...
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darij grinbergDec 23 '11 at 15:38

Daril, you can see the Loday-Ronco Hopf algebra (also called ${\bf PBT}$) as the Hopf subalgebra of ${\bf FQSym}$ generated by all binary trees $T$ where the embedding is defined by $T := \sum_{\sigma \in \mathfrak{S}, {\tt bst}(\sigma) = T} \sigma$, where ${\tt bst}$ is the binary search tree obtained by inserting $\sigma$ (from left to right) following the binary search tree insertion algorithm.
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Samuele GiraudoMar 4 '13 at 20:55

Of course, the homology and cohomology of topological groups over a field are good examples.

For each prime prime $p$ the Steenrod algebra $\mathcal{A}_p$ which is the algebra of endomorphisms of the cohomology theory $H^*(-;\mathbb{F}_p)$. The cohomology of this Hopf algebra is the $E_2$ term of a spectral sequence, due to Adams, converging to the $p$-completed stable homotopy groups of spheres.

The functions on any affine algebraic groups over a field are another family of examples.

Formal group laws over a field $k$. You can read about these in Husemoller's book.

The rational homotopy groups of connected topological group or more generally an $H$-space is a Lie algebra. A nice result of Milnor-Moore shows that the universal enveloping algebra of this Lie algebra is isomorphic as Hopf algebras to the rational homology of the space.

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by
$$\Delta(v) = v \otimes 1 + 1 \otimes v.$$
With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation:
$$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$
while the multiplication is given by the shuffle product:
$$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$
where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e.
$$\sigma(1) < \dots < \sigma(k)$$ and
$$\sigma(k+1) < \dots < \sigma(n).$$
I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.

A couple people mentioned the Steenrod algebra briefly, but you can do a few more topologically-related things:

The subalgebras $\mathcal{A}(n)$ of the Steenrod algebra generated by $Sq^1, ..., Sq^n$ are neat. In particular, it is a good exercise in cohomology to compute $Ext_{\mathcal{A}(n)}(k,k)$. (One can do a minimal resolution and try to look for a pattern, and then prove that it works using a spectral sequence.)

You can show that the Hopf algebras given by $\mathbb{Z}[c_1, c_2, ...]$ ($c_i$ living in degree 2i) and $\mathbb{Z}/2 [w_1, w_2, ...]$ ($w_i$ living in degree i) and comultiplications given by $y_n \mapsto \sum y_i \otimes y_j$ on the generators, are self-dual Hopf algebras and explicitly describe the relationship between itself and the dual. This is neat in and of itself, but then you can mention that these results lead to quick calculations of $H_*MU$ and $H_*MO$ as comodules over the dual of the Steenrod algebra, and thus allow for computations of cobordism groups via the Adams spectral sequence. This self-duality can also be used for a quick proof of the Bott periodicity theorem, though the only reference I know of for this is not yet published (by May), though it will be soon.

If you're doing cohomology, it's always nice to do the cohomology of an exterior algebra; i.e. a Hopf algebra that's an exterior algebra on primitive generators. It's a very easy result, but you can use it to compute other things via spectral sequences.

I know I've forgotten several things I wanted to mention... if I remember them, I'll edit them in.

There's a (unique) semisimple noncocommutative Hopf algebra of dimension 8 that makes a nice example. (Unfortunately I don't remember where to find information on it at the moment, I learned about it in a survey paper of Susan Montgomery's.)

Maybe you are misremembering and have the Sweedler 4-dimensional algebra in mind? I am pretty sure there are many 8-dimensional non-comm, non-cocomm Holp algebras.
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Mariano Suárez-Alvarez♦Dec 23 '11 at 21:00